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oint
oint, v. Obs. or arch. Forms: 4–8 oynt, 6 oynct, 6–9 oint. [f. F. oint, 3 sing. pres. ind., or pa. pple. of oindre:—L. ung(u)ĕre to anoint.] trans. = anoint v.1375 Creation 632 in Horstmann Altengl. Leg. (1878) 132 Of oyle taken ȝow som del, Wherwiþ ȝe mowen oynten me wel. ? a 1400 Cursor M. 7377 (C... Oxford English Dictionary
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oint-plaster
† ˈoint-plaster Obs. In 6 oynt-playster. [Cf. oint v., also OF. oint n.] A plaster of ointment.1578 Lyte Dodoens iii. cxiii. 306 To be applyed, outwardly in oynt-playsters. Oxford English Dictionary
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What situations should $\oint$ be used? Sometimes people put a circle through the integral symbol: $\oint$ What does this mean, and when should we use this integration symbol?
This symbol is used to indicate a line integral along a closed loop. if the loop is the boundary of a compact region $\Omega$ we use also the symbol $ \int_{\delta \Omega} $ we can generalize such notation to the boundary of a region in an n-dimensional space and, if $\Omega$ is an orientable manifo...
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For the positively oriented circle, $\oint_{|z|=1}\frac{2\Re(z)}{z+1}dz=......$ > For the positively oriented circle, $\oint_{|z|=1}\frac{2\Re(z)}{z+1}dz=......$ > > (a) $0$ > > (b) $\pi i$ > > (c) $2\pi i$ > > (d...
By letting $z=e^{i\theta}$ we have $dz= i e^{i\theta}d\theta$ and the wanted integral equals $$ \int_{-\pi}^{\pi}\frac{\text{Re}(e^{i\theta})}{1+e^{i\theta}}ie^{i\theta}\,d\theta =i\int_{-\pi}^{\pi}\frac{\cos\theta(e^{i\theta}+1)}{2+2\cos\theta}\,d\theta=i\color{blue}{\int_{-\pi}^{\pi}\frac{\cos\the...
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From $ \oint \frac{\phi(t)}{t - z} dt = 0$ prove that $ \oint \frac{t \phi(t)}{t - z} dt = 0$ Let $D$ - a simply connected bounded domain and $\phi(t)∈C(\partial D)$. And let $ \displaystyle \oint \frac{\phi(t)}{t - z...
You didn't really say what you mean by $dt$ (some kind of arc length measure, I'm assuming). Ignoring this issue, we have that $$ \int \frac{t\phi(t)}{t-z}\, dt = \int \frac{(t-z+z)\phi(t)}{t-z}\, dt =\int\phi(t)\, dt , $$ so the (holomorphic) function $F(z)=\int\frac{t\phi(t)}{t-z}\, dt$ is constan...
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Why is $\oint_p \frac{1}{z} dz=0$ in simply connected regions? According to Cauchy's Theorem, $\oint_p f(z) dz=0$ for any function $f(z)$ holomorphic on a simply connected region $U$ along any closed-path around any p...
As said in the comments, Cauchy's theorem only applies to simply connected regions. For example, if you choose $U=\mathbb C\setminus \\{0\\}$, then $\frac1z$ is holomorphic in $U$, but $U$ is not simply connected. If $U$ is simply connected and does not contain $0$, then you cannot find a path that ...
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English translation for " tra mi io oint - Translation-Dictionary
tra mi io oint English translation: 透射点.... Please click for detailed translation, meaning, pronunciation and example sentences for tra mi io oint in English www.translation-dictionary.net
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Evaluate $\oint \mathbf{F}\mathbf \cdot\mathbf{n} ds \quad\text{ where }\quad \mathbf F=y \mathbf i+x\mathbf j$ > $$\oint \mathbf{F}\mathbf \cdot\mathbf{n} ds \quad\text{ where }\quad \mathbf F=y \mathbf i+x\mathbf j$...
From what I understand about your problem, you're integrating $$\int\;\vec{F}\cdot d\vec{s} = \iint\;\vec{F}\cdot \vec{n} \; dx dy$$ on a circle of radius 1. So basically the only thing you need to do is to express your vector $d\vec{s}$ in the easiest way. For example, in this case, you're dealing ...
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Compute $\oint_{\partial D_3(0)}\frac{\cos(z+4)}{z^2+1}dz$ I want to compute $$\oint_{\partial D_3(0)}\frac{\cos(z+4)}{z^2+1}dz$$ without using the residue theorem. My plan was to do a partial fractians decomposition:...
Yes, it is fine, except that you forgot to multiply by $2\pi i$ at the last line. The answer is $\pi\bigl(\cos(4+i)-\cos(4-i)\bigr)$.
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Integrate: $\oint_c (x^2 + iy^2)ds$ How do I integrate the following with $|z| = 2$ and $s$ is the arc length? The answer is $8\pi(1+i)$ but I can't seem to get it. $$\oint_c (x^2 + iy^2)ds$$
Use polars such that, for $\theta \in [0,2 \pi)$: $$x = 2 \cos{\theta}$$ $$y = 2 \sin{\theta}$$ $$ds = 2 d\theta$$ That last relation is from the length of an arc of a circle of radius $r$, that is, $ds = r d\theta$.
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How to calculate $\oint\frac{dz}{z^3(z+4)}$ for $|z-2|<3$? Which is right $$\oint\frac{dz}{z^3(z+4)}=2\pi i(\text{Res}(f,0)+\text{Res}(f,-4))$$ or $$\oint\frac{dz}{z^3(z+4)}=2\pi i\,\text{Res}(f,0)$$? I am unsure...
The second is right !!!. Namely, $\displaystyle 2\pi\,\mathrm{i}\,{1 \over 2!}\,\lim_{z \to 0}{\mathrm{d}^{2} \over \mathrm{d}z^{2}}\left[z^{3}\,{1 \over z^{3}\left(z + 4\right)}\right] = \pi\,\mathrm{i}\,\lim_{z \to 0}{2 \over \left(z + 4\right)^{3}} = \bbox[5px,border:1px dotted navy]{\displaystyl...
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Evaluate $\oint x\,dx$ over a particular curve $C$ > $$\oint x\,dx\qquad C:\\{x=0,y=0,y=-x+1 \\} $$ **My attempt:** $$\oint x\,dx=\int_{\uparrow}x\,dx+\int_{\nwarrow}x\,dx+\int_{\rightarrow}x\,dx$$ I don't know wha...
The usual orientation is counter-clockwise so the path $C$, a triangle, would consist of the line segments from $(0,0)$ to $(1,0)$, then from $(1,0)$ to $(0,1)$ and finally from $(0,1)$ back to $(0,0)$. If you really want to explicitly calculate the line integral, you should parametrize each part. W...
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Arnold Trivium 49 : $ \oint_{|z|=2}\frac{dz}{\sqrt{1+z^{10}}}$. Calculate : $$ \oint_{|z|=2}\frac{dz}{\sqrt{1+z^{10}}}.$$ * * * If you find it too easy, then just post hints.
Choosing the squareroot which is positive on the positive real axis we have $${1\over\sqrt{1+z^{10}}}={1\over z^5}(1+z^{-10})^{-1/2}={1\over z^5}\sum_{k=0}^\infty{-1/2\choose k}z^{-10k}\ ,$$ where the binomial series converges uniformly on $\partial D_2$. Therefore $$J:=\int_{\partial D_2}{dz\over\s...
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What is the difference between the line integrals $\oint_b\,ds$, $\oint_b\,dx$, and $\oint_b\,x\,ds$? Can someone explain to me what this means: Where c is the path of a circle with some radius b in clockwise directio...
* The first integral is the arclength of the curve. (Since your curve is a circle of radius $b$, this is just $2\pi b$.) * The second integral is the line integral of the vector field $\langle x,y\rangle=\langle 1,0\rangle$ along the curve. By symmetry, this is zero. * The third integral is the "mas...
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Solving $\oint_C \frac{3z-2}{z^2 - 2iz} \,dz $ using Cauchy's Integral Formula I want to solve the following: $$\oint_C \frac{3z-2}{z^2 - 2iz} \,dz$$ where $C$ is the circle of radius $2$ centered at $z = i$ with a co...
> see that, $\color{blue}{\frac{3z-2}{z^2 - 2iz} = -\frac{i}{z}+\frac{3+i}{z - 2i}}$ By Cauchy formula we get $$\oint_C \frac{3z-2}{z^2 - 2iz} \,dz=-i\ oint_C \frac{dz}{z} \,dz +(3+i)\oint_C \frac{dz}{z - 2i} =\color{red}{2i\pi(-i+3+i)=6i\pi}$$
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